Some Multi-Step Iterative Schemes for Solving Nonlinear Equations

Abstract

Abstract. In this paper, we suggest and analyze a family of multi-step iterative methods which do not involve the high-order differentials of the function for solving nonlinear equations using a different type of decomposition (mainly due to Noor and Noor [15]). We also discuss the convergence of the new proposed methods. Several numerical examples are given to illustrate the efficiency and the performance of the new iterative method. Our results can be considered as an improvement and refinement of the previous results.

Application of homotopy perturbation method to solve a nonlinear mathematical model of depletion of forest resources

“…For this purpose a wide range of methods have been developed to find approximate solutions that are as close as possible to the exact solutions. Among these methods are the Taylor series method [8], which approximates functions as power series; the Picard method [9], which iteratively computes solutions from initial conditions; the Adomian decomposition method [10], which decomposes a differential equation into simpler sub problems; the variational iteration method [11], which uses Lagrange multipliers to optimize solutions; and the homotopy perturbation method [12][13][14]14,15,[17][18][19], which constructs a homotopy that gradually deforms the problem into a simpler one Frontiers in Physics frontiersin.org 04 while adding a perturbation term to the solution. These methods have been applied to a wide range of problems in physics, engineering, various fields and have proven to be highly effective in providing accurate approximations to complex dynamical systems.…”